You can realize it as matrices of the form
$$ \begin{pmatrix} \omega^j & 0 \cr
0 & \omega^{-j} \end{pmatrix} $$
where $\omega $ is a primitive root of unity.
The invariant functions, are functions which satisfy $f(u,v)=f(au+bv, cu+dv)$

The following three functions are invariant $u^n$, $v^n$, $uv$. Set $x=u^n$, $y=v^n$ and $z=uv$. Then they satisfy the follwoing syzygy
$$z^n=xy$$